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arXiv · 1509.01207

General and exact approach to percolation on random graphs

Antoine Allard·Laurent Hébert-Dufresne·Jean-Gabriel Young·Louis J. Dubé
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Abstract

We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite size quenched or random multitype graphs. (ii) We define a very general random graph ensemble that encompasses most of the models published to this day, and also that permits to model structural properties not yet included in a theoretical framework. Site and bond percolation on this ensemble is solved exactly in the infinite size limit using probability generating functions [i.e., the percolation threshold, the size and the composition of the giant (extensive) and small components]. Several examples and applications are also provided. (iii) Our approach can be adapted to model interdependent graphs---whose most striking feature is the emergence of an extensive component via a discontinuous phase transition---in an equally general fashion. We show how a graph can successively undergo a continuous then a discontinuous phase transition, and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition.

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Antoine Allard, Laurent Hébert-Dufresne, Jean-Gabriel Young, Louis J. Dubé. 2015-09-03. General and exact approach to percolation on random graphs. https://doi.org/10.1103/physreve.92.062807

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